Fisher Waves in the Strong Noise Limit

We investigate the effects of a strong number fluctuations on traveling waves in the Fisher-Kolmogorov reaction-diffusion system. Our findings are in stark contrast to the commonly used deterministic and weak-noise approximations. We compute the wave velocity in one and two spatial dimensions, for which we find a linear and a square-root dependence of the speed on the particle density. Instead of smooth sigmoidal wave profiles, we observe fronts composed of a few rugged kinks that diffuse, annihilate, and rarely branch; this dynamics leads to power-law tails in the distribution of the front sizes.

Figure 1

Simulation of the neutral one-dimensional stepping stone model [14]. The model considers a line of sites (demes) labeled by an integer $j$. Each of the demes is occupied by $N$ individuals that carry one of the two genetic variants. Every generation, neighboring demes exchange $Nm$ migrating organisms. Migration is followed by the Wright-Fisher reproduction; i.e., the next generation is formed by $N$ organisms sampled from the Bernoulli distribution with the probability of sampling a particular genetic variant equal to its fraction in the current generation. Here, $N=10$, and $m=0.1$; the total number of demes is 1501. We ensure that the wave front remains within the simulated habitat by moving $\zeta =0$ to the center of the habitat every third generation. The inset shows a snapshot of the wave front $p(t,\zeta )$ [red (lighter) color] and the local heterozygosity $h_0(t,\zeta )$ (solid line) as functions of the deme index $j$ after ${10}^7$ generations starting from the step function initial condition. The main plot shows $\mathcal{F}(\zeta )$ (dashed curve) and $\mathcal{K}(\zeta )$ (solid curve) obtained over ${10}^8$ generations. For large $\zeta$, the functions have the following asymptotic scalings $\mathcal{F}(\zeta )\propto {\zeta }^{-1}$ and $\mathcal{K}(\zeta )\propto {\zeta }^{-2}$ shown by dot-dashed and dotted lines, respectively.

Figure 2

The velocity of planar fronts in the two-dimensional stepping stone model. The simulations are identical to the ones described in Fig. 1 with two exceptions. First, we consider a comoving square lattice of $250\times 250$ sites instead of a linear array. Second, the mutant is $1+S$ times more likely to reproduce than the wild type. The simulations are initialized by a sharp steplike front profile and run for $2\times {10}^7$ generations. The correspondence between the parameters in the simulations and in the SFKPP equation can be expressed as $N=l^2/(b\tau )=10$, $S=a\tau$, $m=D\tau /l^2$, where $l$ and $\tau$ are the lattice constant and the generation time, respectively. For large noise, the wave velocity closely follows a power law $\frac{v}{v_F}\propto (Nm{)}^{1/2}=D/b$. Deviations are primarily due to log corrections and mimic a power law with an exponent slightly smaller than $-1/2$. Note, the scaling regimes almost collapse in a plot $\nu \equiv (v/v_F)\sqrt{s/m}=(v/v_F)\sqrt{a{l}^2/D}$ versus $\beta \equiv (NS{)}^{-1}=b/(a{l}^2)$, see the inset. The solid line is the solution of ${\nu }^2=c_1/[\beta \ln (c_2\nu )]$ with the fitting parameters $c_1=0.45$ and $c_2=1$.